Learn on PengiReveal Math, AcceleratedUnit 4: Sampling and Statistics

Lesson 4-5: Assess Visual Overlap

In this Grade 7 Reveal Math, Accelerated lesson, students learn how to assess visual overlap between two data distributions by comparing their centers using IQR and MAD as measures of variability. Students practice analyzing box plots and histograms to determine whether the difference between medians or means is greater or less than 1 IQR or 2 MADs, which signals how noticeably the distributions differ. Real-world contexts such as rice yield data and adult height distributions are used to draw meaningful conclusions about two populations.

Section 1

Comparing Box Plots

Property

To compare two or more box plots, analyze their measures of center and spread.
Compare the medians to see which data set has a higher central value.
Compare the interquartile ranges (IQRIQR) and overall ranges to determine which data set has greater variability.

Examples

  • Given two box plots for test scores: Class A has a median of 85 and an IQRIQR of 10. Class B has a median of 80 and an IQRIQR of 20. We can conclude Class A performed better on average (higher median) and had more consistent scores (smaller IQRIQR).
  • If Box Plot X has a range of 50 (95−4595 - 45) and Box Plot Y has a range of 30 (80−5080 - 50), the data in Box Plot X is more spread out overall.
  • If the entire box of Plot P is to the right of the entire box of Plot Q, it indicates that the middle 50% of data values in P are all greater than the middle 50% of data values in Q.

Explanation

Comparing box plots allows for a visual comparison of data distributions. By examining the medians, you can quickly compare the centers of the data sets. The length of the boxes (the IQRIQR) and the length of the whiskers (the range) show how spread out or consistent the data are. A shorter box indicates less variability in the middle half of the data, suggesting more consistent values.

Section 2

Comparing Histograms with Equal Intervals

Property

When comparing histograms with equal intervals, follow these steps:
(1) Verify both histograms use identical interval widths and boundaries;
(2) Compare which intervals contain more or fewer values in each distribution;
(3) Describe where most of the data are located and how spread out the data are overall;
(4) Use these observations to make a reasonable comparison between the groups.

Examples

Section 3

Comparing Distributions: Mean and MAD

Property

When comparing two roughly symmetric data distributions, use the mean to compare their centers and the Mean Absolute Deviation (MAD) to compare their variability (spread).

  • Mean: Indicates the typical value of the distribution. A higher mean means the values in that group are generally larger.
  • MAD: Indicates how spread out the data is from the mean. A smaller MAD means the data is more consistent (clustered closer to the mean), while a larger MAD means the data is more spread out.

Section 4

Making Comparative Inferences Using Center and Variability

Property

Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.
Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.

Examples

  • The mean height of basketball players is 80 inches, while soccer players' is 70 inches. If both teams have a MAD of 2 inches, the 10-inch difference is 5 times the MAD, showing a significant height difference.
  • Two schools' average travel time to school is 15 minutes and 18 minutes. If the MAD for both is 5 minutes, the 3-minute difference is small compared to the spread, suggesting the travel times are not very different overall.
  • Class A has a mean exam score of 88 with a MAD of 2. Class B has a mean of 88 with a MAD of 10. While their averages are identical, Class B's scores are far more spread out and less consistent.

Explanation

To compare two groups, look at both their center and spread. A meaningful difference exists if the gap between their means is large compared to their variability (MAD). If the means are close but the MAD is large, the groups overlap a lot.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Comparing Box Plots

Property

To compare two or more box plots, analyze their measures of center and spread.
Compare the medians to see which data set has a higher central value.
Compare the interquartile ranges (IQRIQR) and overall ranges to determine which data set has greater variability.

Examples

  • Given two box plots for test scores: Class A has a median of 85 and an IQRIQR of 10. Class B has a median of 80 and an IQRIQR of 20. We can conclude Class A performed better on average (higher median) and had more consistent scores (smaller IQRIQR).
  • If Box Plot X has a range of 50 (95−4595 - 45) and Box Plot Y has a range of 30 (80−5080 - 50), the data in Box Plot X is more spread out overall.
  • If the entire box of Plot P is to the right of the entire box of Plot Q, it indicates that the middle 50% of data values in P are all greater than the middle 50% of data values in Q.

Explanation

Comparing box plots allows for a visual comparison of data distributions. By examining the medians, you can quickly compare the centers of the data sets. The length of the boxes (the IQRIQR) and the length of the whiskers (the range) show how spread out or consistent the data are. A shorter box indicates less variability in the middle half of the data, suggesting more consistent values.

Section 2

Comparing Histograms with Equal Intervals

Property

When comparing histograms with equal intervals, follow these steps:
(1) Verify both histograms use identical interval widths and boundaries;
(2) Compare which intervals contain more or fewer values in each distribution;
(3) Describe where most of the data are located and how spread out the data are overall;
(4) Use these observations to make a reasonable comparison between the groups.

Examples

Section 3

Comparing Distributions: Mean and MAD

Property

When comparing two roughly symmetric data distributions, use the mean to compare their centers and the Mean Absolute Deviation (MAD) to compare their variability (spread).

  • Mean: Indicates the typical value of the distribution. A higher mean means the values in that group are generally larger.
  • MAD: Indicates how spread out the data is from the mean. A smaller MAD means the data is more consistent (clustered closer to the mean), while a larger MAD means the data is more spread out.

Section 4

Making Comparative Inferences Using Center and Variability

Property

Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.
Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.

Examples

  • The mean height of basketball players is 80 inches, while soccer players' is 70 inches. If both teams have a MAD of 2 inches, the 10-inch difference is 5 times the MAD, showing a significant height difference.
  • Two schools' average travel time to school is 15 minutes and 18 minutes. If the MAD for both is 5 minutes, the 3-minute difference is small compared to the spread, suggesting the travel times are not very different overall.
  • Class A has a mean exam score of 88 with a MAD of 2. Class B has a mean of 88 with a MAD of 10. While their averages are identical, Class B's scores are far more spread out and less consistent.

Explanation

To compare two groups, look at both their center and spread. A meaningful difference exists if the gap between their means is large compared to their variability (MAD). If the means are close but the MAD is large, the groups overlap a lot.